Transfer function of mode-coupled graded-index multimode optical fibers is given theoretically with an algebraic method, which can be applied to arbitrary index profile and distance. Using the transfer function thus obtained, the relation between the transmission characteristics and the mode coupling due to the random bending of the fiber axis is investigated. As a result it is shown that the length-dependent transmission characteristics such as frequency characteristics, coupling-induced loss and bandwidth can be represented in a unified manner using the coupling length defined in this paper. The frequency characteristics is determined dominantly by the ratio of the fiber length to the coupling length. Further, the length dependencies of both the bandwidth and the coupling-induced loss are represented by a normalized form which does not depend upon the index profile. For the distance which is far from the coupling length, the bandwidth is inversely proportional to the square root of the product of the fiber length and the coupling length, and the coupling-induced loss is proportional to the ratio of the fiber length to the coupling length.

A method to estimae the coupling length of the fiber in which modes couple each other due to the random bending of fiber axis is given. The coupling length is estimated from the comparison of the measured transfer function to the theoretical one which is derived previously. The coupling length of a graded-index fiber is estimated to be about 0.8 km using this method. Further, the coupling length of recent graded-index fibers is presumed to be about several Km. For step-index fibers, it is estimated to be in the range between 0.5-5 Km.